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1 Answer
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Thumb: 拇(mŭ)指(zhĭ) or 大拇指 or 大拇哥.
The first two words are common while the last one is less common. You would hear it more in oral speaking, in some regions(I heard of it from one of my relatives in HeBei province).

Watch the shape of "拇". The left part refers to "hand", and the right part is "母“， meaning "mother". I think it gets this name because of the importance of the thumb. I consult it in the dictionary of 《康熙字典》. It cites "【說文】將指也。" I believe "將指" is an analogy that a thumb is like a general among fingers :)

Index Finger:食指(shí zhĭ)

I think we call it because we would dip this finger to the food(soup, etc) to taste the flavour(食 means to eat). There is a famous story from 《左传》 related the index finger and we coin a word "染指" from it(here, 染 means "to dip"). I won't tell you the story here, because it's a little bit long and complicated, sorry. You can search an idiom we get from it "食指大动" to find more info about this story.

That's awesome that the ring finger is called the 无名指. Its name is that it lacks a name... my brain hurts.
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Stumpy Joe PeteJun 22 '12 at 9:19

can the little finger also be called xiao mu zhi?
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user868Jul 23 '12 at 0:38

@mayueyang Yes, you can call it 小拇指. I personally don't like to use that word. Thanks, and I will edit my answer.
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HuangJul 23 '12 at 11:39

Another informal way to call them, as I've heard from persons who came from different part of China, is 大拇指，二拇指，三拇指，四拇指，小拇指。The fact that Sougou Pinyin has these words suggests they are widely known.
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NS.X.Jul 23 '12 at 19:03

Just like Chinese we call ring finger 'without a name finger' in Russia.
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user1174Oct 10 '12 at 5:06